Monotone paths in line arrangement

We show that for any $n$ there is an arrangement of $n$ lines which co ntain an $x$-monotone path of length $\Omega(n^{7/4})$.

[1]  Hiroshi Imai,et al.  Algorithms for vertical and orthogonal L1 linear approximation of points , 1988, SCG '88.

[2]  Tamal K. Dey,et al.  Improved Bounds for Planar k -Sets and Related Problems , 1998, Discret. Comput. Geom..

[3]  Herbert Edelsbrunner,et al.  Algorithms in Combinatorial Geometry , 1987, EATCS Monographs in Theoretical Computer Science.

[4]  János Pach,et al.  Combinatorial Geometry , 2012 .

[5]  Géza Tóth.  Point Sets with Many k-Sets , 2001, Discret. Comput. Geom..

[6]  Leonidas J. Guibas,et al.  Topologically sweeping an arrangement , 1986, STOC '86.

[7]  Jiří Matoušek.  Lower bounds on the length of monotone paths in arrangements , 1991, Discret. Comput. Geom..